Lecture 5: Optical fibers





Optical fiber basics
Linearly polarized modes
Field analysis/wave equation of weakly
guiding fibers
Attenuation in fibers
Dispersion in fibers
References: Photonic Devices, Jia-Ming Liu, Chapter 3
*Most of the lecture materials here are adopted from ELEC342 notes.
1
Optical fiber structure
•  A typical bare fiber consists of a core, a cladding, and a polymer
jacket (buffer coating).
typically
250 µm
including
jacket for
glass fibers
•  The polymer coating is the first line of mechanical protection.
•  The coating also reduces the internal reflection at the cladding, so
light is only guided by the core.
2
Silica optical fibers
•  Both the core and the cladding are made from a type of glass known
as silica (SiO2) which is almost transparent in the visible and near-IR.
•  In the case that the refractive index changes in a “step” between the
core and the cladding. This fiber structure is known as step-index fiber.
•  The higher core refractive index (~ 0.3% higher) is typically achieved
by doping the silica core with germanium dioxide (GeO2).
3
n1
n2
n2
refractive
< 1%
index
125 µm
n2 cladding
n1 core
8 µm
n1 core
125 µm
62.5 µm
n1
n2
n2
Step-index silica optical fiber cross-section
n2 cladding
refractive
< 1%
index
Multi-mode fiber
Single-mode fiber
•  Multi-mode fiber: core dia. ~ 50 or 62.5 or 100 µm; cladding dia. ~ 125 µm
•  Single-mode fiber: core dia. ~ 8 - 9 µm; cladding dia. ~ 125 µm
Both fiber types can have the same numerical aperture (NA) because
NA is independent of the fiber core diameter!
4
Light ray guiding condition
•  Light ray that satisfies total internal reflection at the interface of the
higher refractive index core and the lower refractive index cladding can
be guided along an optical fiber.
cladding n2
core n1
θ
θ
e.g. Under what condition will light be trapped inside the fiber core?
n1 = 1.46; n2 = 1.44
θ > θc
θc = sin-1 (n2/n1) = sin-1 (1.44/1.46) = 80.5o
5
Acceptance angle
•  Only rays with a sufficiently shallow grazing angle (i.e. with an
angle to the normal greater than θc) at the core-cladding interface
are transmitted by total internal reflection.
A
θa
na
n1
n2
αc
θc
•  Ray A incident at the critical angle θc at the core-cladding interface
enters the fiber core at an angle θa to the fiber axis, and is refracted
at the air-core interface.
6
Acceptance angle
θ < θa
θ > θa
θa
θa
na
na
n1
n2
αc
θc
n1
n2
αc
θc
•  Any rays which are incident into the fiber core at an angle > θa
have an incident angle less than θc at the core-cladding interface.
These rays will NOT be totally internal reflected, thus eventually loss
to radiation (at the cladding-jacket interface).
7
Acceptance angle
•  Light rays will be confined inside the fiber core if it is input-coupled
at the fiber core end-face within the acceptance angle θa.
e.g. What is the fiber acceptance angle when n1 = 1.46 and
n2 = 1.44?
θc = sin-1 (n2/n1) = 80.5o => αc = 90o - θc = 9.5o
using sin θa = n1 sin αc
(taking na = 1)
θa = sin-1 (n1 sin αc) = sin-1 (1.46 sin 9.5o) ~ 14o
=> the acceptance angle θa ~ 14o
8
Fiber numerical aperture
In fiber optics, we describe the fiber acceptance angle using
Numerical Aperture (NA):
NA = na sin θa = sin θa = (n12 - n22)1/2
θa
na
n1
n2
αc
θc
•  We can relate the acceptance angle θa and the refractive indices of the
core n1, cladding n2 and air na.
9
Fiber numerical aperture
•  Assuming the end face at the fiber core is flat and normal to the
fiber axis (when the fiber has a “nice” cleave), we consider the
refraction at the air-core interface using Snell’s law:
At θa: na sin θa = n1 sin αc
launching the light from air:
(na ~ 1)
sin θa = n1 sin αc
= n1 cos θc
= n1 (1 - sin2θc)1/2
= n1 (1 - n22/n12)1/2
= (n12 - n22)1/2
10
Fiber numerical aperture
•  Fiber NA therefore characterizes the fiber’s ability to gather light
from a source and guide the light.
e.g. What is the fiber numerical aperture when n1 = 1.46 and
n2 = 1.44?
NA = sin θa = (1.462 - 1.442)1/2 = 0.24
•  It is a common practice to define a relative refractive index Δ as:
Δ = (n1 - n2) / n1
(n1 ~ n2) => NA = n1 (2Δ)1/2
i.e.
Fiber NA only depends on n1 and Δ.
11
Typical fiber NA
•  Silica fibers for long-haul transmission are designed to have
numerical apertures from about 0.1 to 0.3.
•  The low NA makes coupling efficiency tend to be poor, but
turns out to improve the fiber’s bandwidth! (details later)
•  Plastic, rather than glass, fibers are available for short-haul
communications (e.g. within an automobile). These fibers are restricted
to short lengths because of the relatively high attenuation in plastic
materials.
•  Plastic optical fibers (POFs) are designed to have high numerical
apertures (typically, 0.4 – 0.5) to improve coupling efficiency, and so
partially offset the high propagation losses and also enable alignment
tolerance.
12
Meridional and skew rays



A meridional ray is one that has no φ component – it passes
through the z axis, and is thus in direct analogy to a slab
guide ray.
Ray propagation in a fiber is complicated by the possibility of
a path component in the φ direction, from which arises a skew
ray.
Such a ray exhibits a spiral-like path down the core, never
crossing the z axis.
Meridional ray
Skew ray
13
Linearly polarized modes
14
Skew ray decomposition in the core of a step-index fiber
(n1k0)2 = βr2 + βφ2 + β2 = βt2 + β2
15
Vectorial characteristics of modes in optical fibers
•  TE (i.e. Ez = 0) and TM (Hz = 0) modes are also obtained within the
circular optical fiber. These modes correspond to meridional rays
(pass through the fiber axis).
•  As the circular optical fiber is bounded in two dimensions in the
transverse plane,
=> two integers, l and m, are necessary in order to specify the modes
i.e. We refer to these modes as TElm and TMlm modes.
core
cladding
fiber axis
z
x
core
cladding
16
Vectorial characteristics of modes in optical fibers
•  Hybrid modes are modes in which both Ez and Hz are nonzero.
These modes result from skew ray propagation (helical path without
passing through the fiber axis). The modes are denoted as HElm
and EHlm depending on whether the components of H or E make
the larger contribution to the transverse field.
core
cladding
•  The full set of circular optical fiber modes therefore comprises:
TE, TM (meridional rays), HE and EH (skew rays) modes.
17
Weak-guidance approximation
•  The analysis may be simplified when considering telecommunications
grade optical fibers. These fibers have the relative index difference
Δ << 1 (Δ = (ncore – nclad)/ncore typically less than 1 %).
=> the propagation is preferentially along the fiber axis (θ ≈ 90o).
i.e. the field is therefore predominantly transverse.
=> modes are approximated by two linearly polarized components.
(both Ez and Hz are nearly zero)
Δ << 1
core
Two near linearly polarized modes
z
cladding
18
Linearly polarized modes
•  These linearly polarized (LP) modes, designated as LPlm, are
good approximations formed by exact modes TE, TM, HE and EH.
•  The mode subscripts l and m describe the electric field intensity
profile. There are 2l field maxima around the the fiber core
circumference and m field maxima along the fiber core radial direction.
core
fundamental
mode (LP01)
LP21
Electric field
intensity
LP11
LP02
19
Intensity plots for the first six LP modes
LP01
LP02
LP11
LP31
LP21
LP12
20
Plot of the normalized propagation constant b as a
function of V for various LP modes
b
2.405
V
V = (2πa/λ) (n12 – n22)1/2 = (u2 + w2)1/2
b = (β2 – k22)/(k12 – k22)
(see p.41)
21
Number of guided modes
The total number of guided modes M for a step-index fiber is
approximately related to the V number (for V > 20) as follows,
M ≈ V2 / 2
e.g. A multimode step-index fiber with a core diameter of 80 µm and
a relative index difference of 1.5 % is operating at a wavelength of
0.85 µm. If the core refractive index is 1.48, estimate (a) the normalized
frequency for the fiber; (b) the number of guided modes.
(a) V = (2π/λ) a n1 (2Δ)1/2 = 75.8
(b) M ≈ V2 / 2 = 2873 (i.e. nearly 3000 guided modes!)
22
Cutoff wavelength

The cutoff wavelength for any mode is defined as the
maximum wavelength at which that mode propagates. It is
the value of λ that corresponds to Vc for the mode concerns.
For each LP mode, the two parameters are related
λc(lm) = (2πa/(Vc(lm)) (n12 – n22)1/2

The range of wavelengths over which mode lm will propagate
is thus 0 < λ < λc(lm).
For a fiber to operate single mode, the operating wavelength
must be longer than the cutoff wavelength for the LP11 mode.
This is an important specification for a single-mode fiber, and
is usually given the designation λc. We find λc by setting Vc
= 2.405. The range of wavelengths for singlemode operation
is λ > λc.
23
Singlemode condition
For single-mode operation, only the fundamental LP01 mode exists.
The cutoff normalized frequency (Vc) for the next higher order (LP11)
mode in step-index fibers occurs at Vc = 2.405.
=> single-mode propagation of the LP01 mode in step-index fibers:
V < 2.405
e.g. Determine the cutoff wavelength for a step-index fiber to exhibit
single-mode operation when the core refractive index is 1.46 and the core radius is
4.5 µm, with the relative index difference of 0.25 %.
λc = (2πan1/2.405) (2Δ)1/2 = 1214 nm.
Hence, the fiber is single-mode for λ > 1214 nm.
24
Gaussian approximation for the LP01 mode field



The LP01 mode intensity varies with radius as J02(ur/a) inside
the core and as K02(wr/a) in the cladding. The resultant
intensity profile turns out to closely fits a Gaussian function
having a width w0, known as the mode-field radius.
This is defined as the radial distance from the core center to
the 1/e2 point of the Gaussian intensity profile.
A similar Gaussian approximation can be applied to the
fundamental symmetric slab waveguide mode.
E(r) = E(0) exp (-r2 / w02)
=>
I(r) = I(0) exp(-2r2/w02)
Mode-field diameter (MFD) = 2w0 (rather than the core diameter)
characterizes the functional properties of single-mode fibers.
(w0 is also called the spot size.)
25
Mode-field diameter
ncore
“Corning SMF-28” single-mode
fiber has MFD:
nclad
core dia.
9.2 µm at 1310 nm
10.4 µm at 1550 nm
core diameter: 8.2 µm
MFD > core diameter
MFD
26
Mode-field diameter vs. wavelength
11 µm
λc ~ 1270 nm
λ = 1550 nm
λ = 1320 nm
1550 nm
core
•  Mode-field intensity distribution can be measured directly by
near-field imaging the fiber output.
Why characterize the MFD for single-mode fibers?
27
Mode-field diameter mismatch
Ans.: Mismatches in mode-field diameter can increase fiber splice loss.
e.g. Splicing loss due to MFD mismatch between two different
SMF’s
~ dB loss per splice
8 µm
10 µm
SMF1
splicing
SMF2
(A related question: why do manufacturers standardize the
cladding diameter?)
28
Remarks on single-mode fibers
•  no cutoff for the fundamental mode
•  there are in fact two normal modes with orthogonal polarizations
E
E
29
Fiber birefringence



In ideal fibers with perfect rotational symmetry, the two
modes are degenerate with equal propagation constants (βx =
βy), and any polarization state injected into the fiber will
propagate unchanged.
In actual fibers there are imperfections, such as asymmetrical
lateral stresses, non-circular cores and variations in
refractive-index profiles. These imperfections break the
circular symmetry of the ideal fiber and lift the degeneracy of
the two modes.
The modes propagate with different phase velocities, and the
difference between their effective refractive indices is called
the fiber birefringence,
B = |ny – nx|
30
Real optical fiber geometry is by no means perfect
Corning SMF-28 single-mode fiber glass geometry
1. cladding diameter: 125.0 ± 0.7 µm
2. core-cladding concentricity: < 0.5 µm
3. cladding non-circularity: < 1%
[1- (min cladding dia./max clad dia.)]
31
Fiber birefringence
•  State-of-polarization in a constant
birefringent fiber over one
beat length. Input beam is linearly
polarized between the slow
π
and fast axis.
π/2
fast
axis
slow
axis
2π
3π/2
Lbeat = λ / B ~ 1 m
(B ~ 10-6)
*In optical pulses, the polarization state will also be different for
different spectral components of the pulse.
32
Remark on polarizing effects of conventional / polarizationpreserving fibers
Unpol. input
Unknown output
(random coupling
between all the
polarizations present)
Pol. input
Unknown output
Unpol. input
Unknown output
conventional
polarizationpreserving
Pol. Input
(aligned with
a principal axis)
Pol. output
33
Polarization-preserving fibers

The fiber birefringence is enhanced in single-mode
polarization-preserving (polarization-maintaining) fibers,
which are designed to maintain the polarization of the
launched wave.

Polarization is preserved because the two normal modes have
significantly different propagation characteristics. This keeps
them from exchanging energy as they propagate through the
fiber.

Polarization-preserving fibers are constructed by designing
asymmetries into the fiber. Examples include fibers with
elliptical cores (which cause waves polarized along the major
and minor axes of the ellipse to have different effective
refractive indices) and fibers that contain nonsymmetrical
stress-producing parts.
34
Polarization-preserving fibers
Elliptical-core fiber
bow-tie fiber
•  The shaded region in the bow-tie fiber is highly doped with a
material such as boron. Because the thermal expansion of this
doped region is so different from that of the pure silica cladding,
a nonsymmetrical stress is exerted on the core. This produces
a large stress-induced birefringence, which in turn decouples the
two orthogonal modes of the singlemode fiber.
35
Field analysis/wave equation
of weakly guiding fibers
Key derivations for your own reading
36
Field analysis of weakly guiding fibers


Here, we begin the LP mode analysis by assuming field
solutions that are linearly polarized in the fiber transverse
plane.
These consist of an electric field that can be designated as
having x-polarization and a magnetic field that is polarized
along y –
the weak-guidance character of the fiber results in nearly
plane wave behavior for the fields, in which E and H are
orthogonal and exist primarily in the transverse plane (with
very small z components).
E = ax Ex(r, φ, z) = ax Ex0(r, φ) exp (–iβz)
H = ay Hy(r, φ, z) = ay Hy0(r, φ) exp (–iβz)
37
Field analysis of weakly guiding fibers

Because rectangular components are assumed for the fields,
the wave equation
∇t2E0 + (k2 – β2) E0 = 0
is fully separable into the x, y, and z components
∇t2Ex1 + (n12k02 – β2) Ex1 = 0 r ≤ a
∇t2Ex2 + (n22k02 – β2) Ex2 = 0 r ≥ a
where (n12k02 – β2) = βt12 and (n22k02 – β2) = βt22
38
Field analysis of weakly guiding fibers

Assuming transverse variation in both r and φ, we find for the
wave equation, in either region
∂2Ex/∂r2 + (1/r)∂Ex/∂r + (1/r2)∂2Ex/∂φ2 + βt2Ex = 0

We assume that the solution for Ex is a discrete series of
modes, each of which has separated dependences on r, φ, and
z in product form:
Ex = Σ Ri(r)Φi(φ) exp(-iβiz)
i

Each term (mode) in the expansion must itself be a solution
of the wave equation. A single mode, Ex = RΦ exp(-iβz) can
be substituted into the wave equation to obtain
(r2/R) d2R/dr2 + (r/R) dR/dr + r2βt2 = -(1/Φ) d2Φ/dφ2
39
Field analysis of weakly guiding fibers

The left-hand side depends only on r, whereas the right-hand
side depends only on φ.

Because r and φ vary independently, it follows that each side
of the equation must be equal to a constant.

Defining this constant as l2, we can separate the equation into
two equations
d2Φ/dφ2 + l2Φ = 0
d2R/dr2 + (1/r) dR/dr + (βt2 – l2/r2)R = 0


We identify the term l/r as βφ for LP modes.
The bracketed term therefore becomes βt2 - l2/r2 = βr2
40
Solving the Φ wave equation

We can now readily obtain solutions to the Φ equation:
Φ(φ) = cos(lφ + α) or sin (lφ + α)
where α is a constant phase shift.

l must be an integer because the field must be self-consistent
on each rotation of φ through 2π.

The quantity l is known as the angular or azimuthal mode
number for LP modes.
41
Solving the R wave equation

The R-equation is a form of Bessel’s equation. Its solution is
in terms of Bessel functions and assumes the form
R(r) = A Jl(βtr)
= C Kl(|βt|r)
βt real
βt imaginary
where Jl are ordinary Bessel functions of the first kind of
order l, which apply to cases of real βt. If βt is imaginary,
then the solution consists of modified Bessel functions Kl.
42
Bessel functions
LP01
Ordinary Bessel functions
of the first kind
LP01
Modified Bessel functions of
the second kind
•  The ordinary Bessel function Jl is oscillatory, exhibiting no
singularities (appropriate for the field within the core).
•  The modified Bessel function Kl resembles an exponential decay
(appropriate for the field in the cladding).
43
Complete solution for Ex and Hy
•
Define normalized transverse phase / attenuation constants,
u = βt1a = a(n12k02 – β2)1/2
w = |βt2|a = a(β2 – n22k02)1/2

Using the cos(lφ) dependence (with constant phase shift α = 0),
we obtain the complete solution for Ex:
Ex = A Jl(ur/a) cos (lφ) exp(-iβz)
Ex = C Kl(wr/a) cos (lφ) exp(-iβz)

r≤a
r≥a
Similarly, we can solve the wave equation for Hy
Hy = B Jl(ur/a) cos (lφ) exp(-iβz)
r≤a
Hy = D Kl(wr/a) cos (lφ) exp(-iβz) r ≥ a
where A ≈ Z B and C ≈ Z D in the quasi-plane-wave
approximation, and Z ≈ Z0/n1 ≈ Z0/n2
44
Electric field for LPlm modes

Applying the field boundary conditions at the core-cladding
interface:
Eφ1|r=a = Eφ2|r=a
n12Er1|r=a = n22Er2|r=a
Hφ1|r=a = Hφ2|r=a
µ1Hr1|r=a = µ2Hr2|r=a
where µ1 = µ2 = µ0, Hr1|r=a = Hr2|r=a.
•  In the weak-guidance approximation, n1 ≈ n2, so Er1|r=a ≈ Er2|r=a
⇒ Ex1|r=a ≈ Ex2|r=a
Hy1|r=a ≈ Hy2|r=a
•  Suppose A = E0,
Ex = E0 Jl(ur/a) cos (lφ) exp (-iβz)
(r ≤ a)
Ex = E0 [Jl(u)/Kl(w)] Kl(wr/a) cos (lφ) exp (–iβz) (r ≥ a)
45
Electric fields of the fundamental mode

The fundamental mode LP01 has l = 0 (assumed x-polarized)
Ex = E0 J0(ur/a) exp (-iβz)
(r ≤ a)
Ex = E0 [J0(u)/K0(w)] K0(wr/a) exp (–iβz) (r ≥ a)
These fields are cylindrically symmetrical, i.e. there is no
variation of the field in the angular direction.
They approximate a Gaussian distribution. (see the J0(x)
distribution on p. 40)
46
Intensity patterns

The LP modes are observed as intensity patterns.

Analytically we evaluate the time-average Poynting vector
|<S>| = (1/2Z) |Ex|2
Defining the peak intensity I0 = (1/2Z) |E0|2, we find the
intensity functions in the core and cladding for any LP mode
Ilm = I0 Jl2(ur/a) cos2(lφ)
r≤a
Ilm = I0 (Jl(u)/Kl(w))2 Kl2(wr/a) cos2(lφ)
r≥a
47
Eigenvalue equation for LP modes

We use the requirement for continuity of the z components of the
fields at r = a
Hz = (i/ωµ) (∇ x E)z
⇒
•
(∇ x E1)z|r=a = (∇ x E2)z|r=a
Convert E into cylindrical components
E1 = E0 Jl(ur/a) cos(lφ) (arcos φ – aφsin φ) exp (-iβz)
E2 = E0 [Jl(u)/Kl(w)] Kl(wr/a) cos(lφ) (arcos φ – aφsin φ) exp(–iβz)
48
Eigenvalue equation for LP modes
•  Taking the curl of E1 and E2 in cylindrical coordinates:
(∇ x E1)z = (E0/r) {[lJl(ur/a) – (ur/a)Jl-1(ur/a)] cos (lφ) sin φ
+ lJl(ur/a) sin (lφ) cos φ}
(∇ x E2)z = (E0/r)(Jl(u)/Kl(u)){[lKl(wr/a)–(wr/a)Kl-1(wr/a)] cos lφ sin φ
+ lKl(wr/a) sin (lφ) cos φ}
where we have used the derivative forms of Bessel functions.
•  Using (∇ x E1)z|r=a = (∇ x E2)z|r=a
uJl-1(u)/Jl(u) = -w Kl-1(w)/Kl(w)
This is the eigenvalue equation for LP modes in the step-index
fiber.
49
Cutoff condition

Cutoff for a given mode can be determined directly from the
eigenvalue equation by setting w = 0 (see p.41),
u = V = Vc
(Recall from p.21 V2 = u2 + w2)
where Vc is the cutoff (or minimum) value of V for the mode of
interest.

The cutoff condition according to the eigenvalue equation is
VcJl-1(Vc)/Jl(Vc) = 0
When Vc ≠ 0, Jl-1(Vc) = 0
e.g. Vc = 2.405 as the cutoff value of V for the LP11 mode.
50
Attenuation in fibers
51
Transmission characteristics of optical fibers
•  The transmission characteristics of most interest: attenuation (loss)
and bandwidth.
•  Now, silica-based glass fibers have losses less than 0.2 dB/km (i.e.
95 % launched power remains after 1 km of fiber transmission). This
is essentially the fundamental lower limit for attenuation in silicabased glass fibers.
•  Fiber bandwidth is limited by the signal dispersion within the fiber.
Bandwidth determines the number of bits of information transmitted
in a given time period. Now, fiber bandwidth has reached many 10’s
Gbit over many km’s per wavelength channel.
52
Attenuation
•  Signal attenuation within optical fibers is usually expressed in the
logarithmic unit of the decibel.
The decibel, which is used for comparing two power levels, may be
defined for a particular optical wavelength as the ratio of the
output optical power Po from the fiber to the input optical power Pi.
Loss (dB) = - 10 log10 (Po/Pi) = 10 log10 (Pi/Po)
(Po ≤ Pi)
*In electronics, dB = 20 log10 (Vo/Vi)
53
Attenuation in dB/km
*The logarithmic unit has the advantage that the operations of
multiplication (and division) reduce to addition (and subtraction).
Po/Pi = 10[-Loss(dB)/10]
In numerical values:
The attenuation is usually expressed in decibels per unit length
(i.e. dB/km):
γ L = - 10 log10 (Po/Pi)
γ (dB/km): signal attenuation per unit length in decibels
L (km): fiber length
54
dBm
•  dBm is a specific unit of power in decibels when the reference power
is 1 mW:
dBm = 10 log10 (Power/1 mW)
e.g. 1 mW = 0 dBm; 10 mW = 10 dBm; 100 µW = - 10 dBm
=>
Loss (dB) = input power (dBm) - output power (dBm)
e.g. Input power = 1 mW (0 dBm), output power = 100 µW (-10 dBm)
⇒ loss = -10 log10 (100 µW/1 mW) = 10 dB
OR 0 dBm – (-10 dBm) = 10 dB
55
Fiber attenuation mechanisms
1.
2.
3.
4.
5.
Material absorption
Scattering loss
Nonlinear loss
Bending loss
Mode coupling loss
•  Material absorption is a loss mechanism related to both the material
composition and the fabrication process for the fiber. The optical
power is lost as heat in the fiber.
•  The light absorption can be intrinsic (due to the material components
of the glass) or extrinsic (due to impurities introduced into the glass
during fabrication).
56
Intrinsic absorption
•  Pure silica-based glass has two major intrinsic absorption
mechanisms at optical wavelengths:
(1) a fundamental UV absorption edge, the peaks are centered in the
ultraviolet wavelength region. This is due to the electron transitions
within the glass molecules. The tail of this peak may extend into the
the shorter wavelengths of the fiber transmission spectral window.
(2)  A fundamental infrared and far-infrared absorption edge,
due to molecular vibrations (such as Si-O). The tail of these absorption
peaks may extend into the longer wavelengths of the fiber transmission
spectral window.
57
Fundamental fiber attenuation characteristics
IR absorption
UV absorption
(negligible in the IR)
58
Electronic and molecular absorption

Electronic absorption: the bandgap of fused silica is about
8.9 eV (~140 nm). This causes strong absorption of light in
the UV spectral region due to electronic transitions across the
bandgap.
In practice, the bandgap of a material is not sharply defined
but usually has bandtails extending from the conduction and
valence bands into the bandgap due to a variety of reasons,
such as thermal vibrations of the lattice ions and microscopic
imperfections of the material structure.
An amorphous material like fused silica generally has very
long bandtails. These bandtails lead to an absorption tail
extending into the visible and infrared regions. Empirically,
the absorption tail at photon energies below the bandgap falls
off exponentially with photon energy.
59
Electronic and molecular absorption

Molecular absorption: in the infrared region, the absorption
of photons is accompanied by transitions between different
vibrational modes of silica molecules.

The fundamental vibrational transition of fused silica causes
a very strong absorption peak at about 9 µm wavelength.

Nonlinear effects contribute to important harmonics and
combination frequencies corresponding to minor absorption
peaks at 4.4, 3.8 and 3.2 µm wavelengths.
=> a long absorption tail extending into the near infrared,
causing a sharp rise in absorption at optical wavelengths
longer than 1.6 µm.
60
Extrinsic absorption
•  Major extrinsic loss mechanism is caused by absorption due to
water (as the hydroxyl or OH- ions) introduced in the glass fiber during
fiber pulling by means of oxyhydrogen flame.
•  These OH- ions are bonded into the glass structure and have
absorption peaks (due to molecular vibrations) at 1.39 µm. The
fundamental vibration of the OH- ions appear at 2.73 µm.
•  Since these OH- absorption peaks are sharply peaked, narrow spectral
windows exist around 1.3 µm and 1.55 µm which are essentially
unaffected by OH- absorption.
•  The lowest attenuation for typical silica-based fibers occur at
wavelength 1.55 µm at about 0.2 dB/km, approaching the minimum
possible attenuation at this wavelength.
61
1400nm OH- absorption peak and spectral windows
OH- absorption (1400 nm)
(Lucent 1998)
62
Impurity absorption




Impurity absorption: most impurity ions such as OH-, Fe2+
and Cu2+ form absorption bands in the near infrared region
where both electronic and molecular absorption losses of the
host silica glass are very low.
Near the peaks of the impurity absorption bands, an impurity
concentration as low as one part per billion can contribute to
an absorption loss as high as 1 dB km-1.
In fact, fiber-optic communications were not considered
possible until it was realized in 1966 (Kao) that most losses in
earlier fibers were caused by impurity absorption and then
ultra-pure fibers were produced in the early 1970s (Corning).
Today, impurities in fibers have been reduced to levels where
losses associated with their absorption are negligible, with the
exception of the OH- radical.
63
Three major fiber transmission spectral windows
The 1st window: 850 nm, attenuation 4 dB/km
The 2nd window: 1300 nm, attenuation 0.5 dB/km
The 3rd window: 1550 nm, attenuation 0.2 dB/km
1550 nm window is today’s standard long-haul communication
wavelengths.
Short
S band
1460
Conventional Long
C band
1530
1500
Ultra-long
L band
1565
U band
1625
1600
1675
λ (nm)
64
Scattering loss
Scattering results in attenuation (in the form of radiation) as the
scattered light may not continue to satisfy the total internal reflection
in the fiber core.
One major type of scattering is known as Rayleigh scattering.
θ > θc
θ < θc
local point-like
inhomogeneities core
cladding
The scattered ray can escape by refraction according to Snell’s Law.
65
Rayleigh scattering
•  Rayleigh scattering results from random inhomogeneities that are small
in size compared with the wavelength.
<<
λ
•  These inhomogeneities exist in the form of refractive index fluctuations
which are frozen into the amorphous glass fiber upon fiber pulling. Such
fluctuations always exist and cannot be avoided !
Rayleigh scattering results in an attenuation (dB/km) ∝ 1/λ4
Where else do we see Rayleigh scattering?
66
Rayleigh scattering



The intrinsic Rayleigh scattering in a fiber is caused by
variations in density and composition that are built into the
fiber during the manufacturing process. They are primarily a
result of thermal fluctuations in the density of silica glass and
variations in the concentration of dopants before silica passes
its glass transition point to become a solid.
These variations are a fundamental thermodynamic
phenomenon and cannot be completely removed. They create
microscopic fluctuations in the index of refraction, which
scatter light in the same manner as microscopic fluctuations of
the density of air scatter sunlight.
This elastic Rayleigh scattering process creates a loss given by
n: Index of refraction
kB: Boltzmann constant
T: glass transition temperature
β: isothermal compressibility
67
Rayleigh scattering is the dominant loss in today’s fibers
Rayleigh
Scattering (1/λ4)
0.2 dB/km
68
Waveguide scattering

Imperfections in the waveguide structure of a fiber, such as
nonuniformity in the size and shape of the core, perturbations
in the core-cladding boundary, and defects in the core or
cladding, can be generated in the manufacturing process.

Environmentally induced effects, such as stress and
temperature variations, also cause imperfections.

The imperfections in a fiber waveguide result in additional
scattering losses. They can also induce coupling between
different guided modes.
69
Nonlinear losses




Because light is confined over long distances in an optical
fiber, nonlinear optical effects can become important even at
a relatively moderate optical power.
Nonlinear optical processes such as stimulated Brillouin
scattering and stimulated Raman scattering can cause
significant attenuation in the power of an optical signal.
Other nonlinear processes can induce mode mixing or
frequency shift, all contributing to the loss of a particular
guided mode at a particular frequency.
Nonlinear effects are intensity dependent, and thus they can
become very important at high optical powers.
70
Fiber bending loss and mode-coupling to higher-order modes
Phase velocity
cannot exceed c,
and thus radiation
“macrobending”
(how do we measure bending loss?)
“microbending” – power
coupling to higher-order
modes that are more lossy.
71
Dispersion in fibers
72
Dispersion in fibers





Dispersion is the primary cause of limitation on the optical
signal transmission bandwidth through an optical fiber.
Recall from Lecture 4 that there are waveguide and modal
dispersions in an optical waveguide in addition to material
dispersion.
Both material dispersion and waveguide dispersion are
examples of chromatic dispersion because both are frequency
dependent.
Waveguide dispersion is caused by frequency dependence of
the propagation constant β of a specific mode due to the
waveguiding effect. (recall the b vs. V plot of a specific
mode)
The combined effect of material and waveguide dispersions
for a particular mode alone is called intramode dispersion.
73
Modal dispersion



Modal dispersion is caused by the variation in propagation
constant between different modes; it is also called intermode
dispersion. (recall the b vs. V plot at a fixed V)
Modal dispersion appears only when more than one mode is
excited in a multimode fiber. It exists even when chromatic
dispersion disappears.
If only one mode is excited in a fiber, only intramode
chromatic dispersion has to be considered even when the
fiber is a multimode fiber.
74
Material dispersion




For optical fibers, the materials of interest are pure silica and
doped silica.
The parameters of interest are the refractive index n, the
group index ng and the group velocity dispersion D.
The index of refraction of pure silica in the wavelength range
between 200 nm and 4 µm is given by the following
empirically fitted Sellmeier equation:
where λ is in micrometers.
The index of refraction can be changed by adding dopants to
silica, thus facilitating the means to control the index profile
of a fiber. Doping with germania (GeO2) or alumina
increases the index of refraction.
75
Fiber dispersion
•  Fiber dispersion results in optical pulse broadening and hence
digital signal degradation.
Optical pulse
broadened pulse
Optical fiber
input
output
Dispersion mechanisms: 1. Modal (or intermodal) dispersion
2. Chromatic dispersion (CD)
3. Polarization mode dispersion (PMD)
76
Pulse broadening limits fiber bandwidth (data rate)
1 0 1
1 0 1
Intersymbol interference
(ISI)
Signal distorted
Fiber length (km)
•  An increasing number of errors may be encountered on the digital
optical channel as the ISI becomes more pronounced.
77
Modal dispersion
•  When numerous waveguide modes are propagating, they all travel
with different velocities with respect to the waveguide axis.
•  An input waveform distorts during propagation because its energy
is distributed among several modes, each traveling at a different speed.
•  Parts of the wave arrive at the output before other parts, spreading out
the waveform. This is thus known as multimode (modal) dispersion.
•  Multimode dispersion does not depend on the source linewidth
(even a single wavelength can be simultaneously carried by multiple
modes in a waveguide).
•  Multimode dispersion would not occur if the waveguide allows only
78
one mode to propagate - the advantage of single-mode waveguides!
Modal dispersion as shown from the mode chart of a
symmetric slab waveguide
Normalized guide index b
(neff = n1)
m=0
1
TE
2
3
TM
4
5
(neff = n2)
V (∝ 1/λ)
•  Phase velocity for mode m = ω/βm = ω/(neff(m) k0)
(note that m = 0 mode is the slowest mode)
79
Modal dispersion in multimode waveguides
m=2
m=1
m=0
θ2
θ1
θ0
The carrier wave can propagate along all these different “zig-zag”
ray paths of different path lengths.
80
Modal dispersion as shown from the LP mode chart of a silica
optical fiber
Normalized guide index b
(neff = n1)
(neff = n2)
V (∝ 1/λ)
•  Phase velocity for LP mode = ω/βlm = ω/(neff(lm) k0)
(note that LP01 mode is the slowest mode)
81
Modal dispersion results in pulse broadening
fastest mode
m=3
T
Optical pulse
3
T
multimode fiber
2
1
m=0
m=2
T
m=1
T
slowest mode
T
m=0
time
Τ + ΔT
modal dispersion: different modes arrive at the receiver with different
delays => pulse broadening
82
Estimate modal dispersion pulse broadening using phase velocity
•  A zero-order mode traveling near the waveguide axis needs time:
t0 = L/vm=0 ≈ Ln1/c
(vm=0 ≈ c/n1)
n1
L
•  The highest-order mode traveling near the critical angle needs time:
tm = L/vm ≈ Ln2/c
(vm ≈ c/n2)
∼θc
=> the pulse broadening due to modal dispersion:
ΔT ≈ t0 – tm ≈ (L/c) (n1 – n2)
≈ (L/2cn1) NA2
(n1 ~ n2)
83
How does modal dispersion restricts fiber bit rate?
e.g. How much will a light pulse spread after traveling along
1 km of a step-index fiber whose NA = 0.275 and ncore = 1.487?
Suppose we transmit at a low bit rate of 10 Mb/s
=> Pulse duration = 1 / 107 s = 100 ns
Using the above e.g., each pulse will spread up to ≈ 100 ns (i.e. ≈
pulse duration !) every km
⇒ The broadened pulses overlap! (Intersymbol interference (ISI))
*Modal dispersion limits the bit rate of a km-length fiber-optic link to
~ 10 Mb/s. (a coaxial cable supports this bit rate easily!)
84
Bit-rate distance product
•  We can relate the pulse broadening ΔT to the information-carrying
capacity of the fiber measured through the bit rate B.
•  Although a precise relation between B and ΔT depends on many
details, such as the pulse shape, it is intuitively clear that ΔT should
be less than the allocated bit time slot given by 1/B.
⇒ An order-of-magnitude estimate of the supported bit rate is obtained
from the condition BΔT < 1.
⇒ Bit-rate distance product (limited by modal dispersion)
BL < 2c ncore / NA2
This condition provides a rough estimate of a fundamental limitation
of step-index multimode fibers. (smaller the NA larger the bit-rate
85
distance product)
Bit-rate distance product
The capacity of optical communications systems is frequently
measured in terms of the bit rate-distance product.
e.g. If a system is capable of transmitting 100 Mb/s over a distance
of 1 km, it is said to have a bit rate-distance product of
100 (Mb/s)-km.
This may be suitable for some local-area networks (LANs).
Note that the same system can transmit 1 Gb/s along 100 m, or
10 Gb/s along 10 m, or 100 Gb/s along 1 m, or 1 Tb/s along 10 cm,
...
86
Single-mode fiber eliminates modal dispersion
cladding
core
θ0
•  The main advantage of single-mode fibers is to propagate only one
mode so that modal dispersion is absent.
•  However, pulse broadening does not disappear altogether. The group
velocity associated with the fundamental mode is frequency dependent
within the pulse spectral linewidth because of chromatic dispersion. 87
Chromatic dispersion
•  Chromatic dispersion (CD) may occur in all types of optical
fiber. The optical pulse broadening results from the finite spectral
linewidth of the optical source.
intensity 1.0
0.5
Δλ
linewidth
λο
λ(nm)
*In the case of the semiconductor laser Δλ corresponds to only a
fraction of % of the centre wavelength λo. For LEDs, Δλ is
likely to be a significant percentage of λo.
88
Spectral linewidth
•  Real sources emit over a range of wavelengths. This range is the
source linewidth or spectral width.
•  The smaller is the linewidth, the smaller the spread in wavelengths
or frequencies, the more coherent is the source.
•  A perfectly coherent source emits light at a single wavelength.
It has zero linewidth and is perfectly monochromatic.
Light sources
Linewidth (nm)
Light-emitting diodes
20 nm – 100 nm
Semiconductor laser diodes
Nd:YAG solid-state lasers
1 nm – 5 nm
0.1 nm
HeNe gas lasers
0.002 nm
89
Chromatic dispersion
input pulse
broadened pulse
single mode
L
λο+(Δλ/2)
λο
λο-(Δλ/2)
arrives
first
Different spectral components
have different time delays
=> pulse broadening
time
arrives
last
time
•  Pulse broadening occurs because there may be propagation delay
differences among the spectral components of the transmitted signal.
•  Chromatic dispersion (CD): Different spectral components of a pulse
travel at different group velocities. This is also known as group velocity
dispersion (GVD).
90
Light pulse in a dispersive medium
When a light pulse with a spread in frequency δω and a spread in
propagation constant δk propagates in a dispersive medium n(λ),
the group velocity:
vg = (dω/dk) = (dλ/dk) (dω/dλ)
k = n(λ) 2π/λ
=>
ω = 2πc/λ
=>
dω/dλ = -2πc/λ2
Hence
dk/dλ = (2π/λ) [(dn/dλ) - (n/λ)]
vg = c / [n – λ(dn/dλ)] = c / ng
Define the group refractive index ng = n – λ(dn/dλ)
91
group refractive index ng
Group refractive index ng vs. λ for fused silica
n(λ)
ng(λ)
Wavelength (nm)
92
velocity (m/s)
Phase velocity c/n and group velocity c/ng vs. λ for fused silica
Phase velocity
dispersion
v = c/n
vg= c/ng
Group velocity
dispersion (GVD)
Wavelength (nm)
93
Group-Velocity Dispersion (GVD)
Consider a single mode fiber of length L
•  A specific spectral component at the frequency ω (or wavelength λ)
would arrive at the output end of the fiber after a time delay:
T = L/vg
•  If Δλ is the spectral width of an optical pulse, the extent of pulse
broadening for a fiber of length L is given by
ΔT = (dT/dλ) Δλ = [d(L/vg)/dλ] Δλ
= L [d(1/vg)/dλ] Δλ
94
Group-Velocity Dispersion (GVD)
Hence the pulse broadening due to a differential time delay:
ΔT = L D Δλ
where D = d(1/vg)/dλ is called the dispersion parameter and is
expressed in units of ps/(km-nm).
D = d(1/vg)/dλ = c-1 dng/dλ = c-1 d[n – λ(dn/dλ)]/dλ
= -c-1 λ d2n/dλ2
95
Dispersion (ps/km-nm)
Dispersion parameter
Fused silica
D = - (λ/c) d2n/dλ2
1276 nm
“Anomalous”
(D > 0)
Wavelength (nm)
“Normal”
(D < 0)
96
Variation of vg with wavelength for fused silica
vg
“Normal” “Anomalous”
(D > 0)
(D < 0)
Red goes faster Red goes slower
Dmat = 0
@ 1276 nm
Wavelength (nm)
C band
97
Zero-dispersion wavelength
Material dispersion Dmat = 0 at λ ~ 1276 nm for fused silica.
This λ is referred to as the zero-dispersion wavelength λZD.
Chromatic (or material) dispersion D(λ) can be zero;
or
negative => longer wavelengths travel faster than shorter wavelengths;
or
positive => shorter wavelengths travel faster than longer wavelengths.
98
Waveguide dispersion
In fact there are two mechanisms for chromatic dispersion:
(a) Silica refractive index n is wavelength dependent (i.e. n = n(λ))
=> different wavelength components travel at different speeds in silica
This is known as material dispersion.
(b) Light energy of a mode propagates partly in the core and partly in
the cladding. The mode power distribution between the core and
the cladding depends on λ. (Recall the mode field diameter)
This is known as waveguide dispersion.
=> D(λ) = Dmat(λ) + Dwg(λ)
99
Waveguide dispersion in a single-mode fiber
input pulse
MFD
ncore
nclad
core pulse
slower
cladding pulse
faster
Singlemode fiber
time
=>
broadened pulse !
Waveguide dispersion depends on the mode field distribution
in the core and the cladding. (i.e. the fiber V number)
100
Normalized guide index b
Waveguide dispersion of the LP01 mode
LP01
2.405
V (∝ 1/λ)
•  Different wavelength components λ of the LP01 mode see
different effective indices neff
101
Waveguide group velocity and time delay
•  Consider an optical pulse of linewidth Δλ (Δω) and a corresponding
spread of propagation constant Δβ propagating in a waveguide
Group velocity
vg,eff = dω/dβ
vg,eff-1 = dβ/dω
or
= d/dω (c-1 ω neff)
= c-1 (neff + ω dneff/dω)
= c-1 (neff - λ dneff/dλ) = c-1 ng,eff
Time delay after a waveguide of length L: τ = L/vg,eff
Or time delay per unit length: τ/L = vg,eff-1
102
Waveguide dispersion parameter
•  If Δλ is the spectral width of an optical pulse, the extent of pulse
broadening for a waveguide of length L is given by
Δτ = (dτ/dλ) Δλ = [d(L/vg,eff)/dλ] Δλ
= L [d(1/vg,eff)/dλ] Δλ
= L Dwg Δλ
•  Dwg = d(1/vg,eff)/dλ is called the waveguide dispersion
parameter and is expressed in units of ps/(km-nm).
Dwg = d(1/vg,eff)/dλ = c-1 dng,eff/dλ = c-1 d[neff – λ dneff/dλ]/dλ
= -c-1 λ d2neff/dλ2
103
Waveguide dispersion parameter
•  Recall vg,eff = (dβ/dω)-1 and note that the propagation constant
β is a nonlinear function of the V number, V = (2πa/λ) NA = a (ω/c) NA
•  In the absence of material dispersion (i.e. when NA is independent of
ω), V is directly proportional to ω, so that
1/vg,eff = dβ/dω = (dβ/dV) (dV/dω) = (dβ/dV) (a NA/c)
•  The pulse broadening associated with a source of spectral width Δλ is
related to the time delay L/vg,eff by ΔT = L |Dwg| Δλ. The waveguide
dispersion parameter Dwg is given by
Dwg = d/dλ (1/vg,eff) = -(ω/λ) d/dω (1/vg,eff) = -(1/(2πc)) V2 d2β/dV2
⇒ The dependence of Dwg on λ may be controlled by altering the core
radius, the NA, or the V number.
104
Silica fiber dispersion
Typical values of D are about
15 - 18 ps/(km-nm) near 1.55 µm.
Dmat = 0
D = Dmat + Dwg
D=0
λo ~ 1310 nm
•  Dwg(λ) compensate
some of the Dmat(λ) and
shifts the λZD from about
1276 nm to a longer
wavelength of ~1310 nm.
105
Chromatic dispersion in low-bit-rate systems
•  Recall broadening of the light pulse due to chromatic dispersion:
ΔT = D L Δλ
Consider the maximum pulse broadening equals to the bit time
period 1/B, then the dispersion-limited distance:
LD = 1 / (D B Δλ)
e.g. For D = 17 ps/(km•nm), B = 2.5 Gb/s and Δλ = 0.03 nm
=> LD = 784 km
(It is known that dispersion limits a 2.5 Gbit/s channel to roughly
900 km! Therefore, chromatic dispersion is not much of an issue in
106
low-bit-rate systems deployed in the early 90’s!)
Chromatic dispersion scales with B2
•  When upgrading from 2.5- to 10-Gbit/s systems, most technical
challenges are less than four times as complicated and the cost
of components is usually much less than four times as expensive.
•  However, when increasing the bit rate by a factor of 4, the effect of
chromatic dispersion increases by a factor of 16!
•  Consider again the dispersion-limited distance:
LD = 1 / (D B Δλ)
Note that spectral width Δλ is proportional to the modulation of the
lightwave!
i.e. Faster the modulation, more the frequency content, and therefore
wider the spectral bandwidth => Δλ ∝ Β
=> LD ∝ 1 / B2
107
Chromatic dispersion in high-bit-rate systems
e.g. In standard single-mode fibers for which D = 17 ps/(nm•km)
at a signal wavelength of 1550 nm (assuming from the same light
source as the earlier example of 2.5 Gbit/s systems), the maximum
transmission distance before significant pulse broadening occurs
for 10 Gbit/s data is:
LD ~ 784 km / 16 ~ 50 km!
(A more exact calculation shows that 10-Gbit/s (40-Gbit/s) would
be limited to approximately 60 km (4 km!).)
This is why chromatic dispersion compensation must be employed
for systems operating at 10 Gbit/s (now at 40 Gbit/s and beyond.)
108
Zero-dispersion slope
If D(λ) is zero at a specific λ = λZD, can we eliminate
pulse broadening caused by chromatic dispersion?
There are higher order effects! The derivative
dD(λ)/dλ = So
needs to be accounted for when the first order effect is zero
(i.e. D(λZD) = 0) .
So is known as the zero-dispersion slope measured in ps/(km-nm2).
109
Pulse broadening near zero-dispersion wavelength
Dispersion (ps/nm-km)
The chromatic pulse broadening near λZD: ΔT = L So |λ – λZD| Δλ
For Corning SMF-28 fiber, λZD = 1313 nm,
So = 0.086 ps/nm2-km
1313 nm
D(λ) < 0
D(λ) > 0
Wavelength (nm)
empirical D(λ) = (So/4) (λ - λZD4/λ3)
110
Limiting bit rate near zero-dispersion wavelength
•  Now it becomes clear that at λ = λZD, the dispersion slope So becomes
the bit rate limiting factor. We can estimate the limiting bit rate by noting
that for a source of spectral width Δλ, the effective value of dispersion
parameter becomes
D = So Δλ
=>
The limiting bit rate-distance product can be given as
BL |So| (Δλ)2 < 1
(B ΔT < 1)
*For a multimode semiconductor laser with Δλ = 2 nm and a dispersionshifted fiber with So = 0.05 ps/(km-nm2) at λ = 1.55 µm, the BL product
approaches 5 (Tb/s)-km. Further improvement is possible by using
single-mode semiconductor lasers.
111
Dispersion tailored fibers
1.  Since the waveguide contribution Dwg depends on the fiber
parameters such as the core radius a and the index difference Δ, it is
possible to design the fiber such that λZD is shifted into the
neighborhood of 1.55 µm. Such fibers are called dispersion-shifted
fibers.
2. It is also possible to tailor the waveguide contribution such that the
total dispersion D is relatively small over a wide wavelength range
extending from 1.3 to 1.6 µm. Such fibers are called dispersionflattened fibers.
112
Dispersion-shifted and flattened fibers
(standard)
ncore(r)
ncore(r)
•  The design of dispersion-modified fibers often involves the use of
multiple cladding layers and a tailoring of the refractive index profile.113
Non-zero dispersion shifted fibers
Dispersion [ps/nm-km]
•  Since dispersion slope S > 0 for singlemode fibers => different
wavelength-division multiplexed (WDM) channels have different
dispersion values.
WDM
1500
1550 1600
*SM fiber or non-zero
dispersion-shifted
fiber (NZDSF) with
D ~ few ps/(km-nm)
λ
*In fact, for WDM systems, small amount of chromatic dispersion
is desirable in order to prevent the impairment of fiber nonlinearity
114
(i.e. power-dependent interaction between wavelength channels.)
Chromatic Dispersion Compensation
•  Chromatic dispersion is time independent in a passive optical link
⇒ allow compensation along the entire fiber span
(Note that recent developments focus on reconfigurable optical links,
which makes chromatic dispersion time dependent!)
Two basic techniques: (1) dispersion-compensating fiber DCF
(2) dispersion-compensating fiber grating
•  The basic idea for DCF: the positive dispersion in a conventional
fiber (say ~ 17 ps/(km-nm) in the 1550 nm window) can be
compensated for by inserting a fiber with negative dispersion (i.e.
with large -ve Dwg).
115
Chromatic dispersion accumulates linearly over distance
Accumulated dispersion (ps/nm)
(recall ΔT = D L Δλ )
+D (red goes
slower)
time
Positive dispersion
transmission fiber
time
Distance (km)
116
Chromatic Dispersion Compensation
Accumulated dispersion (ps/nm)
Positive dispersion transmission
fiber
-D’
+D
-D’
Negative dispersion element
-D’
+D
-D’
-D’
+D
-D’
Distance (km)
•  In a dispersion-managed system, positive dispersion transmission
fiber alternates with negative dispersion compensation elements,
117
such that the total dispersion is zero end-to-end.
Dispersion [ps/nm-km]
Fixed (passive) dispersion compensation
SM
+ve
17
+ve
SM fiber
λ
λο
DCF
DCF
-ve
SM
+ve
-80
DCF
-ve
-ve (due to large -ve Dwg)
SM
+ve
DCF
-ve
*DCF is a length of fiber producing -ve dispersion four to five times
118
as large as that produced by conventional SMF.
Dispersion-Compensating Fiber
The concept: using a span of fiber to compress an initially chirped pulse.
Pulse broadening with chirping
λl
λs
λl
Pulse compression with dechirping
λs
λl
λs
λs
λl
Initial chirp and broadening by a transmission link Compress the pulse to initial width
L1
L2
Dispersion compensated channel: D2 L2 = - D1 L1
119
Dispersion-Compensating Fiber
Laser
Conventional fiber (D > 0)
DCF (D < 0)
L
LDCF
Detector
e.g. What DCF is needed in order to compensate for dispersion in a
conventional single-mode fiber link of 100 km?
Suppose we are using Corning SMF-28 fiber,
=> the dispersion parameter D(1550 nm) ~ 17 ps/(km-nm)
⇒  Pulse broadening ΔTchrom = D(λ) Δλ L ~ 17 x 1 x 100 = 1700 ps.
assume the semiconductor (diode) laser linewidth Δλ ~ 1 nm.
120
Dispersion-Compensating Fiber
⇒ The DCF needed to compensate for 1700 ps with a large
negative-dispersion parameter
i.e. we need ΔTchrom + ΔTDCF = 0
⇒ ΔTDCF = DDCF(λ) Δλ LDCF
suppose typical ratio of L/LDCF ~ 6 – 7, we assume LDCF = 15 km
=>DDCF(λ) ~ -113 ps/(km-nm)
*Typically, only one wavelength can be compensated exactly.
Better CD compensation requires both dispersion and dispersion
slope compensation.
121
Dispersion slope compensation
Compensating the dispersion slope produces the additional requirement:
L2 dD2/dλ = - L1 dD1/dλ
⇒  The compensating fiber must have a negative dispersion slope, and that the
dispersion and slope values need to be compensated for a given length.
D2 L2 = - D1 L1
L2 dD2/dλ = - L1 dD1/dλ
=> Dispersion and slope compensation: D2 / (dD2/dλ) = D1 / (dD1/dλ)
(In practice, two fibers are used, one of which has negative slope, in which the pulse
wavelength is at zero-dispersion wavelength λzD.)
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Dispersion [ps/nm-km]
Dispersion slope compensation
SM fiber
16
λ1
DCF
17 18
λ2
+ve
λ
-ve (due to large -ve Dwg)
-96
-102
-108
Within the spectral window (λ1, λ2), DDCF/DSM = -6
SDCF = -12/(λ2 – λ1); SSM = 2/(λ2 – λ1) => SDCF/SSM = -6
=> Dispersion slope compensation: (DSM/SSM) / (DDCF/SDCF) = 1
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Disadvantages in using DCF
•  Added loss associated with the increased fiber span
•  Nonlinear optic effects may degrade the signal over the long length of the
fiber if the signal is of sufficient intensity.
•  Links that use DCF often require an amplifier stage to compensate the added
loss.
-D
Splice loss
g
-D
g
Long DCF (loss, possible nonlinear optic effects)
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Polarization Mode Dispersion (PMD)
•  In a single-mode optical fiber, the optical signal is carried by the
linearly polarized “fundamental mode” LP01, which has
two polarization components that are orthogonal.
(note that x and y
are chosen arbitrarily)
•  In a real fiber (i.e. ngx ≠ ngy), the two orthogonal polarization
modes propagate at different group velocities, resulting in pulse
broadening – polarization mode dispersion.
125
Polarization Mode Dispersion (PMD)
Ey ΔT
Ey
vgy = c/ngy
Ex
vgx = c/ngx ≠ vgy
t
Ex
t
Single-mode fiber L km
1. Pulse broadening due to the orthogonal polarization modes
(The time delay between the two polarization components is
characterized as the differential group delay (DGD).)
2. Polarization varies along the fiber length
126
Polarization Mode Dispersion (PMD)
•  The refractive index difference is known as birefringence.
B = nx - ny
(~ 10-6 - 10-5 for
single-mode fibers)
assuming nx > ny => y is the fast axis, x is the slow axis.
*B varies randomly because of thermal and mechanical stresses over
time (due to randomly varying environmental factors in submarine,
terrestrial, aerial, and buried fiber cables).
=> PMD is a statistical process !
127
Randomly varying birefringence along the fiber
y
E
Principal axes
x
Elliptical polarization
128
Randomly varying birefringence along the fiber
•  The polarization state of light propagating in fibers with randomly
varying birefringence will generally be elliptical and would quickly
reach a state of arbitrary polarization.
*However, the final polarization state is not of concern for most
lightwave systems as photodetectors are insensitive to the state of
polarization.
(Note: the recent revival of technology developments in “Coherent
Optical Communications” do require polarization state to be analyzed!)
•  A simple model of PMD divides the fiber into a large number of
segments. Both the magnitude of birefringence B and the orientation
of the principal axes remain constant in each section but changes
randomly from section to section.
129
A simple model of PMD
B0
Lo
B1
L1
B2
B3
L2
L3
Ex
Ey
t
Randomly changing differential group delay (DGD)
130
PMD pulse broadening
•  Pulse broadening caused by a random change of fiber polarization
properties is known as polarization mode dispersion (PMD).
PMD pulse broadening
ΔTPMD = DPMD √L
DPMD is the PMD parameter (coefficient) measured in ps/√km.
√L models the “random” nature (like “random walk”)
* DPMD does not depend on wavelength (first order) ;
*Today’s fiber (since 90’s) PMD parameter is 0.1 - 0.5 ps/√km.
(Legacy fibers deployed in the 80’s have DPMD > 0.8 ps/√km.)
131
PMD pulse broadening
e.g. Calculate the pulse broadening caused by PMD for a singlemode
fiber with a PMD parameter DPMD ~ 0.5 ps/√km and a fiber length of
100 km. (i.e. ΔTPMD = 5 ps)
Recall that pulse broadening due to chromatic dispersion for a 1 nm
linewidth light source was ~ 15 ps/km, which resulted in 1500 ps for
100 km of fiber length.
=> PMD pulse broadening is two orders of magnitude less than
chromatic dispersion !
*PMD is relatively small compared with chromatic dispersion. But
when one operates at zero-dispersion wavelength (or dispersion
compensated wavelengths) with narrow spectral width, PMD can
become a significant component of the total dispersion.
132
So why do we care about PMD?
Recall that chromatic dispersion can be compensated to ~ 0,
(at least for single wavelengths, namely, by designing proper
-ve waveguide dispersion)
but there is no simple way to eliminate PMD completely.
=> It is PMD that limits the fiber bandwidth after chromatic
dispersion is compensated!
133
PMD in 40Gbit/s systems
•  PMD is of lesser concern in lower data rate systems. At lower
transmission speeds (up to and including 10 Gb/s), networks have
higher tolerances to all types of dispersion, including PMD.
As data rates increase, this dispersion tolerance reduces significantly,
creating a need to control PMD as much as possible at the current
40 Gb/s system.
e.g. The pulse broadening caused by PMD for a singlemode
fiber with a PMD parameter of 0.5 ps/√km and a fiber length of
100 km => 5 ps.
However, this is comparable to the 40G bit period = 25 ps !
134